Let $K=\frac{\mathbb{Q}[x]}{<f(x)>}$ where $f(x)$ is irreducible over $\mathbb{Q}$ and has even degree. I want to find $K_2$ such that $ \mathbb{Q} \subseteq K_2\subseteq K$ and $[K_2:\mathbb{Q}]=2$.
If K is a cyclic Galois extension over $\mathbb{Q}$ then discriminant of $f(x)$ solves the problem.
But what if $K$ is not a cyclic galois extension?
Or even particular when $K$ is not even a Galois extension.